76 PRINCIPLES OF SYMBOLICAL REASONING. [CHAP. V. 



stituent involves x as a factor, change in the original function x 

 into 1 ; but if it involves 1 - x as a factor, change in the original 

 function x into 0. Apply the same rule with reference to the 

 symbols y, z, &c. : the final calculated value of the function thus 

 transformed will be the coefficient sought. 



The sum of the constituents, multiplied each by its respective 

 coefficient, will be the expansion required. 



12. It is worthy of observation, that a function may be de- 

 veloped with reference to symbols which it does not explicitly 

 contain. Thus if, proceeding according to the rule, we seek to 

 develop the function 1 - a, with reference to the symbols x and 

 y, we have, 



When x = 1 and y = 1 the given function = 0. 

 x= 1 y = =0. 



z = y = 1 =1. 



z = y = =1. 



Whence the development is 



1 - x = xy + x (1 - y) + (1 - #) y + (1 - x) (1 - y) ; 

 and this is a true development. The addition of the terms ( 1 - x)y 

 and (1 - x) (1 - y) produces the function 1 - x. 



The symbol 1 thus developed according to the rule, with re- 

 spect to the symbol #, gives 



x + I - x. 

 Developed with respect to x and y, it gives 



xy + x (1 - y) + (1 - x) y + (1 - x) (1 - y). 



Similarly developed with respect to any set of symbols, it pro- 

 duces a series consisting of all possible constituents of those 

 symbols. 



13. A few additional remarks concerning the nature of the 

 general expansions may with propriety be added. Let us take, 

 for illustration, the general theorem (5), which presents the type 

 of development for functions of two logical symbols. 



In the first place, that theorem is perfectly true and intel- 

 ligible when x and y are quantitative symbols of the species con- 

 sidered in this chapter, whatever algebraic form may be assigned 

 to the function /(#, ?/), and it may therefore be intelligibly em- 



